I flew SW and used DME last August. No problems whatsoever. But, no, SW is not one of the 'participating' airlines. This impacts only your return to MCO from your resort. With SW, you will just take your bags with you to the DME bus. The driver will put the bags underneath the bus and you have to check them, as usual, at the airport. Those using 'participating' airlines can check their bags right at their resort and not have to worry about taking their bags to the airport.
The baggage claim area is different than the DME desk area. I would put the yellow tags on the bags and just bypass the baggage claim area. Then, when you get off the plane, just keep going downwards until you get to the bottom, where you will find DME, in the same area as the car rental people.
The only time I would suggest taking the bags from baggage claim (and not putting the yellow tags on beforehand) is if you are arriving late in the day. That way you would have your bags with you for the night.